# Location–scale family

In probability theory, especially in mathematical statistics, a location–scale family is a family of probability distributions parametrized by a location parameter and a non-negative scale parameter. For any random variable $X$ whose probability distribution function belongs to such a family, the distribution function of $Y{\stackrel {=}}a+bX$ also belongs to the family (where ${\stackrel {=}}$ means "equal in distribution"—that is, "has the same distribution as"). Moreover, if $X$ and $Y$ are two random variables whose distribution functions are members of the family, and assuming

1. existence of the first two moments and
2. $X$ has zero mean and unit variance,

then $Y$ can be written as $Y{\stackrel {=}}\mu _+\sigma _X$ , where $\mu _$ and $\sigma _$ are the mean and standard deviation of $Y$ .

In other words, a class $\Omega$ of probability distributions is a location–scale family if for all cumulative distribution functions $F\in \Omega$ and any real numbers $a\in \mathbb$ and $b>0$ , the distribution function $G(x)=F(a+bx)$ is also a member of $\Omega$ .

• If $X$ has a cumulative distribution function $F_(x)=P(X\leq x)$ , then $Y{=}a+bX$ has a cumulative distribution function $F_(y)=F_\left({\frac }\right)$ .
• If $X$ is a discrete random variable with probability mass function $p_(x)=P(X=x)$ , then $Y{=}a+bX$ is a discrete random variable with probability mass function $p_(y)=p_\left({\frac }\right)$ .
• If $X$ is a continuous random variable with probability density function $f_(x)$ , then $Y{=}a+bX$ is a continuous random variable with probability density function $f_(y)={\frac }f_\left({\frac }\right)$ .

In decision theory, if all alternative distributions available to a decision-maker are in the same location–scale family, and the first two moments are finite, then a two-moment decision model can apply, and decision-making can be framed in terms of the means and the variances of the distributions.

## Examples

Often, location–scale families are restricted to those where all members have the same functional form. Most location–scale families are univariate, though not all. Well-known families in which the functional form of the distribution is consistent throughout the family include the following:

## Converting a single distribution to a location–scale family

The following shows how to implement a location–scale family in a statistical package or programming environment where only functions for the "standard" version of a distribution are available. It is designed for R but should generalize to any language and library.

The example here is of the Student's t-distribution, which is normally provided in R only in its standard form, with a single degrees of freedom parameter df. The versions below with _ls appended show how to generalize this to a generalized Student's t-distribution with an arbitrary location parameter mu and scale parameter sigma.

 Probability density function (PDF): dt_ls(x, df, mu, sigma) = 1/sigma * dt((x - mu)/sigma, df) Cumulative distribution function (CDF): pt_ls(x, df, mu, sigma) = pt((x - mu)/sigma, df) Quantile function (inverse CDF): qt_ls(prob, df, mu, sigma) = qt(prob, df)*sigma + mu Generate a random variate: rt_ls(df, mu, sigma) = rt(df)*sigma + mu

Note that the generalized functions do not have standard deviation sigma since the standard t distribution does not have standard deviation of 1.